\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 213, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/213\hfil Pr\"ufer substitutions]
{Pr\"ufer substitutions on a coupled system involving the $p$-Laplacian}
\author[W.-C. Wang\hfil EJDE-2013/213\hfilneg]
{Wei-Chuan Wang} % in alphabetical order
\address{Wei-Chuan Wang \newline
Center for General Education, National Quemoy
University, Kinmen, 892, Taiwan}
\email{wangwc72@gmail.com}
\thanks{Submitted July 12, 2013. Published September 25, 2013.}
\subjclass[2000]{34A55, 34B24, 47A75}
\keywords{Coupled system; $p$-Laplacian; Pr\"ufer substitution}
\begin{abstract}
In this article, we employ a modified Pr\"ufer substitution acting
on a coupled system involving one-dimensional $p$-Laplacian equations.
The basic properties for the initial valued problem and some estimates are
obtained. We also derive an analogous Sturmian theory and give a
reconstruction formula for the potential function.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks
\section{Introduction}
There has been recently a lot of interest in the study of the
$p$-Laplacian eigenvalue problem
\begin{gather*}
-\Delta_py+q|y|^{p-2}y=\lambda |y|^{p-2}y,\\
y|_{\partial \Omega}=0,
\end{gather*}
where $p>1$ and $q\in C(\Omega)$, $\Omega \subseteq \mathbb{R}^n$.
This is a quasilinear partial differential equation when $p\neq 2$.
The most cited application is the highly viscid fluid flow
(cf. Ladyzhenskaya \cite{la} and Lions \cite{li}).
When $p=2$, $q$ and $\lambda$ both vanish, it becomes the Laplacian equation.
The $p$-Laplacian operator has the originally physical meaning, and can also
be treated as a generalization of the Laplacian operator.
For the one-dimensional case,
the $p$-Laplacian eigenvalue problem becomes, after scaling,
\begin{gather}
\label{eq1.4}
-(y'^{(p-1)})'=(p-1)(\lambda -q(x))y^{(p-1)}, \\
\label{eq1.5}
y(0)=y(1)=0,
\end{gather}
where $p>1$, $f^{(p-1)}\equiv |f|^{p-1}\operatorname{sgn} f=|f|^{p-2}f$, and $q$
is a continuous function defined on $[0,1]$. The following
Sturm-Liouville property for the one-dimensional $p$-Laplacian
operator is well-known now (cf. Binding \& Drabek \cite{bd},
Reichel \& Walter \cite{rw99}, Walter \cite{W98}, etc.).
\begin{theorem} \label{thm1.1}
For \eqref{eq1.4}-\eqref{eq1.5}, there exists a sequence
of eigenvalues $\{\lambda_n\}_{n=1}^{\infty}$ such that
$$
-\infty1$, and $q$ is a
continuous function defined in $\mathbb{R}$. When $p=2$,
\eqref{eq1.1} reduces to
\begin{equation}\label{eq1.3}
\begin{gathered}
u''(x)+\lambda u(x)-q(x)v(x)=0,\\
v''(x)+\lambda v(x)+q(x)u(x)=0,
\end{gathered}
\end{equation}
which is a linear coupled system. One can treat \eqref{eq1.3} as a
steady state reaction diffusion model.
Define $H(u,v)=\frac{\lambda}{2}u^2-\frac{\lambda}{2}v^2-q(x)uv$.
Then
$$
\frac{\partial H}{\partial u}=\lambda u-q(x)v,\quad
-\frac{\partial H}{\partial v}=\lambda v+q(x)u.
$$
Equation \eqref{eq1.3} can be viewed as
a simplest model of diffusion systems with skew-gradient
structure (cf. \cite{Y02,Y021}).
Here we intend to study the existence of sign-changing solutions
(or nodal solutions) of \eqref{eq1.1}-\eqref{eq1.2} and try to
derive an analog of Theorem \ref{thm1.1}. Employing the
information of solutions, a reconstruction formula for $q(x)$ is given.
Such a procedure is called an inverse nodal problem.
An inverse problem of this type was designated by McLaughlin \cite{M88} in 1988.
When one studies the inverse nodal problem of
\eqref{eq1.1}-\eqref{eq1.2}, an interesting observation arises.
The asymptotic formula given in Theorem \ref{thm1.3} (see the following)
coincides with the one of the classical Sturm-Liouville eigenvalue problem
\begin{gather*}
-y''+w_0(x) y=\mu y,\\
y(0)=y(1)=0
\end{gather*}
(cf. \cite{M88,S88,LSY}). Besides, the Pr\"ufer substitution is
an efficient method in showing the oscillation property for
solutions (cf. \cite{BB}). In this article we utilize a modified
Pr\"ufer substitution to treat this problem. Fortunately we can
tackle the effect of the two coupled functions in
\eqref{eq1.1}-\eqref{eq1.2}, and obtain the detailed estimates of
parameters $\lambda_m$ and nodal points. The following are our main results.
\begin{theorem} \label{thm1.2}
There exists a sequence of real parameters
$\{\lambda_k\}_{k=m}^{\infty}$ of the one-dimensional coupled
system \eqref{eq1.1}-\eqref{eq1.2}, where $m\in \mathbb{N}$ such that
$$
00$, the amplitude functions satisfy that
\begin{equation} \label{eq2.16}
2\exp[-c_1\lambda^{\frac{1-p}{p}}x]\leq R(x,\lambda)^{p-1}+r(x,\lambda)^{p-1}\leq
2\exp[c_2\lambda^{\frac{1-p}{p}}x],
\end{equation}
where $c_1,~c_2$ are some
positive constants.
(ii) For fixed $x>0$ and sufficiently large $\lambda$, we have
\begin{equation} \label{eq2.17}
\frac{r(x,\lambda)}{R(x,\lambda)}=1+o(1).
\end{equation}
Moreover, $\frac{R(x,\lambda)}{r(x,\lambda)}$ has the same
asymptotic estimate as in \eqref{eq2.17}.
\end{lemma}
\begin{proof}
(i) By assumption and \eqref{eq2.12} and \eqref{eq2.14},
there exist some positive constants $c_1$ and $c_2$ such that
\begin{align*}
&-c_1\lambda^{\frac{1-p}{p}}[R(x)^{p-1}+r(x)^{p-1}]\\
&\leq R(x)^{p-2}R'(x)+r(x)^{p-2}r'(x)\leq
c_2\lambda^{\frac{1-p}{p}}[R(x)^{p-1}+r(x)^{p-1}].
\end{align*}
Solving the above differential inequality and applying the initial condition
\eqref{eq2.15}, we obtain the inequality \eqref{eq2.16}.
(ii) As in $(i)$, there exists some positive
constant $c_3$ such that
$$
\frac{R(x)r'(x)-r(x)R'(x)}{R(x)^2}\leq
c_3\lambda^{\frac{1-p}{p}}[\frac{R(x)^{p-2}}{r(x)^{p-2}}+\frac{r(x)^p}{R(x)^p}].
$$
Letting $y(x)=\frac{r(x)}{R(x)}$, we have
$$
y'(x)\leq c_3\lambda^{\frac{1-p}{p}}[y(x)^{2-p}+y(x)^p].
$$
Note that
$$
\frac{dy}{dx}\leq c_3\lambda^{\frac{1-p}{p}}(\frac{1+y^{2p-2}}{y^{p-2}});\quad
\text{i.e., }
\frac{y^{p-2}dy}{1+y^{2p-2}}\leq c_3\lambda^{\frac{1-p}{p}}dx.
$$
Let $z=y^{p-1}$ and integrate the
above inequality; we obtain
$$
\tan^{-1}(y(x)^{p-1})-\tan^{-1}(y(0)^{p-1})\leq
(p-1)c_3\lambda^{\frac{1-p}{p}}x;
$$
i.e.,
$$
0< \tan^{-1}(y(x)^{p-1})\leq \tan^{-1}(1)+(p-1)c_3\lambda^{\frac{1-p}{p}}x.
$$
So
\begin{equation} \label{eq2.18}
y(x)^{p-1}\leq 1+o(1)
\end{equation}
as $\lambda$ is sufficiently large. This completes the proof.
\end{proof}
From Proposition \ref{prop2.2} and \eqref{eq2.16}, we have
the following property.
\begin{proposition} \label{prop2.5}
For any fixed $\lambda\in \mathbb{R}^+$, problem \eqref{eq1.1}-\eqref{eq1.2}
has a unique solution which exists over the whole interval $[0,1]$.
\end{proposition}
\section{The Sturmian property}
In this section, we first derive the following lemma for the proof of
Theorem \ref{thm1.2}.
\begin{lemma} \label{lem3.1}
For $\lambda>0$, the phase angle function $\theta(x,\lambda)$
satisfies the following properties.
\begin{itemize}
\item[(i)] $\theta(\cdot,\lambda)$ is continuous in $\lambda$
and satisfies $\theta(0,\lambda)=0$.
\item[(ii)] If $\lambda^{1/p}\theta(x_n,\lambda)=n\pi_p$ for some
$x_n\in (0,1)$, then $\lambda^{1/p}\theta(x,\lambda)>n\pi_p$ for every $x>x_n$.
\item[(iii)] \begin{equation}
\label{eq3.2}
\lim_{\lambda \to \infty}\lambda^{1/p}\theta(1,\lambda)=\infty.
\end{equation}
\end{itemize}
\end{lemma}
\begin{proof}
For (i), $\theta(\cdot,\lambda)$ is continuous in $\lambda$ followed by
the continuous dependence on parameters.
And $\theta(0,\lambda)=0$ is valid by \eqref{eq2.15}.
Also if
$\lambda^{1/p}\theta(x_n,\lambda)=n\pi_p$ for some
$x_n\in (0,1)$, then by \eqref{eq2.11} and Lemma \ref{lem2.4}, we have
\begin{equation} \label{eq3.1}
\theta'(x_n,\lambda)=1>0.
\end{equation}
This proves (ii). For (iii), integrating \eqref{eq2.11} over
$[0,1]$ and applying (i), one obtains
\begin{equation} \label{eq3.3}
\lambda^{1/p}\theta(1,\lambda)
=\lambda^{1/p}-\lambda^{\frac{1-p}{p}}\int_0^1q(t)
(\frac{r(t,\lambda)}{R(t,\lambda)})^{p-1}S_p
(\lambda^{1/p}\theta(t,\lambda))S_p(\lambda^{1/p}\phi(t,\lambda))^{(p-1)}dt.
\end{equation}
By \eqref{eq2.17}, one has
$$\lambda^{1/p}\theta(1,\lambda)=\lambda^{1/p}+O(\frac{1}{\lambda^{1-\frac{1}{p}}})$$
for sufficiently large $\lambda$. This completes the proof.
\end{proof}
We remark that using \eqref{eq2.13} and Lemma \ref{lem2.4},
one can apply the similar arguments as in the above proof to obtain
the conclusions in Lemma \ref{lem3.1} for
the phase function $\phi(x,\lambda)$.
\begin{proof}[Proof of Theorem \ref{thm1.2}]
By Lemma \ref{lem3.1}, for every sufficiently large
$k\in\mathbb{N}$, there exists $\lambda_k >0$ satisfies
$\lambda_k^{1/p}\theta(1,\lambda_k)=k\pi_p$. This implies that
there exists $m\in\mathbb{N}$ such that
$\lambda_k^{1/p}\theta(1,\lambda_k)=k\pi_p$ for every $k\geq m$.
In this case, $\lambda_m < \lambda_{m+1}